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WHAT IS GOING ON?

The Days (And Years) Of Our Lives

Looking at the numbers above, you'll immediately notice that you are different ages on the different planets. This brings up the question of how we define the time intervals we measure. What is a day? What is a year?

The earth is in motion. Actually, several different motions all at once. There are two that specifically interest us. First, the earth rotates on its axis, like a spinning top. Second, the earth revolves around the sun, like a tetherball at the end of a string going around the center pole.

The top-like rotation of the earth on its axis is how we define the day. The time it takes the earth to rotate from noon until the next noon we define as one day. We further divide this period of time into 24 hours, each of which is divided into 60 minutes, each of which is broken into 60 seconds. There are no rules that govern the rotation rates of the planets, it all depends on how much "spin" was in the original material that went into forming each one. Giant Jupiter has lots of spin, turning once on its axis every 10 hours, while Venus takes 243 days to spin once.

The revolution of the earth around the sun is how we define the year. A year is the time it takes the earth to make one revolution - a little over 365 days.

We all learn in grade school that the planets move at differing rates around the sun. While earth takes 365 days to make one circuit, the closest planet, Mercury, takes only 88 days. Poor, ponderous, and distant Pluto takes a whopping 248 years for one revolution. Below is a table with the rotation rates and revolution rates of all the planets.

 

Why the huge differences in periods? We need to go back to the time of Galileo, except that we're not going to look at his work, but rather at the work of one of his contemporaries, Johannes Kepler (1571-1630).

Kepler briefly worked with the great Danish observational astronomer, Tycho Brahe. Tycho was a great and extremely accurate observer, but he did't have the mathematical capacity to analyze all of the data he collected. After Tycho's death in 1601, Kepler was able to obtain Tycho's observations. Tycho's observations of planetary motion were the most accurate of the time (before the invention of the telescope!). Using these observations, Kepler discovered that the planets do not move in circles, as 2000 years of "Natural Philosophy" had taught. He discovered that they move in ellipses. A ellipse is a sort of squashed circle with a short diameter (the "minor axis") and a longer diameter (the "major axis"). He found that the Sun was positioned at one "focus" of the ellipse (there are two "foci", both located on the major axis). He also found that when the planets were nearer the sun in their orbits, they move faster than when they were farther from the sun. Many years later, he discovered that the farther a planet was from the sun, on the average, the longer it took for that planet to make one complete revolution. These three laws, stated mathematically by Kepler, are known as "Kepler's Laws of Orbital Motion." Kepler's Laws are still used today to predict the motions of planets, comets, asteroids, stars, galaxies, and spacecraft.

 

 

Here you see a planet in a very elliptical orbit.
Note how it speeds up when it's near the Sun.

Kepler's third law is the one that interests us the most. It states precisely that the period of time a planet takes to go around the sun squared is proportional to the average distance from the sun cubed. Here's the formula:

 

Period²=Distance³

Let's just solve for the period by taking the square root of both sides:

Period=√ Distance³ 

Note that as the distance of the planet from the sun is increased, the period, or time to make one orbit, will get longer. Kepler didn't know the reason for these laws, though he knew it had something to do with the Sun and its influence on the planets. That had to wait 50 years for Isaac Newton to discover the universal law of gravitation.

 

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The Gravity Of The Situation

Closer planets revolve faster, more distant planets revolve slower. Why? The answer lies in how gravity works. The force of gravity is a measure of the pull between two bodies. This force depends on a few things. First, it depends on the mass of the sun and on the mass of the planet you are considering. The heavier the planet, the stronger the pull. If you double the planet's mass, gravity pulls twice as hard. On the other hand, the farther the planet is from the sun, the weaker the pull between the two. The force gets weaker quite rapidly. If you double the distance, the force is one-fourth. If you triple the separation, the force drops to one-ninth. Ten times the distance, one-hundredth the force. See the pattern? The force drops off with the square of the distance. If we put this into an equation it would look like this:

The two "M's" on top are the sun's mass and the planet's mass. The "r" below is the distance between the two. The masses are in the numerator because the force gets bigger if they get bigger. The distance is in the denominator because the force gets smaller when the distance gets bigger. Note that the force never becomes zero no matter how far you travel. Knowing this law helps you understand why the planets move faster when they are closer to the sun - they are pulled on with a stronger force and are whipped around faster!

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